I want to calculate the poisson bracket of two angular momentum components: $$\{L^i,L^j\}=\frac{\partial\epsilon^{\:i\:m}_{\:\:l}q^lp_m}{\partial q^k} \frac{\partial\epsilon_{\:\:s}^{\:j\:r}q^sp_r}{\partial p_k}-\frac{\partial\epsilon^{\:i\:m}_{\:\:l}q^lp_m}{\partial p_k} \frac{\partial\epsilon_ {\:\:s}^{\:j\:r} q^sp_r}{\partial q^k}$$ Which gives: $$\{L^i,L^j\}= \epsilon^{\:i\:m}_{\:\:k} \epsilon_{\:\:s}^{\:j\:k}p_mq^s- \epsilon^{\:i\:k}_{\:\:l} \epsilon_{\:\:k}^{\:j\:r}p_rq^l $$ But then I get problems with the index positions if I want to convert the Levi-Civita-Tensors into Kronecker-Deltas. Is angular momentum defined different in general coordinate systems or where else is the problem?
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